Week 06.04.2009 – 12.04.2009

We consider quivers/skew-symmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all the vertices, and give families of quivers which have higher periodicity. The periodicity means that sequences given by recurrence relations arise in a natural way from the associated cluster algebras. We present a number of interesting, new nonlinear recurrences, necessarily with the Laurent property, of both the real line and the plane. In particular, we show that some of these recurrences can be linearised and, with certain initial conditions, give integer sequences which contain all solutions of some particular Pell equations.

It is well established numerically that the spectral statistics of pseudo-integrable models differ considerably from the reference statistics of integrable and chaotic systems. In a previous paper by Bogomolny and Schmit the statistical properties of a certain quantized pseudo-integrable map had been calculated analytically but only for a special sequence of matrix dimensions. This talk aims at describing the method in order to obtain the spectral statistics of the same quantum map for all matrix dimensions.